Kinetic equation for classical particles obeying an exclusion principle

Abstract

In this paper we analyze the kinetics of classical particles which obey an exclusion principle (EP) in the only-individual-transitions (OIT) approximation, and separately in the more rigorous contemporary-transitions (CT) description. In order to be able to include the EP into the kinetics equations we consider a discrete, one-dimensional, heterogeneous and anisotropic phase space and, after defining the reduced transition probabilities, we write a master equation. As a limit to the continuum of this master equation we obtain a generalized Fokker-Planck (FP) equation. This last is a nonlinear partial differential equation and reduces to the standard FP equation if the nonlinear term, which takes into account the EP, is neglected. The steady states of this equation, both in the OIT approximation and CT description, are considered. In the particularly interesting case of Brownian particles as a steady state in the OIT approximation we obtain the Fermi-Dirac (FD) distribution, while in the CT description we obtain another distribution which differs slightly from that of the FD. Moreover, our approach permits us to treat in an alternative and efficient way the problem of the determination of an effective potential to simulate the exclusion principle in classical many-body equations of motion.